We consider S 2 bundles P and P ′ of totally null planes of maximal dimension over a 4-dimensional manifold equipped with a Weyl or Riemannian geometry. The fibre product PP ′ of P and P ′ is found to be appropriate for the encoding of both the selfdual and the Einstein-Weyl equations for the 4-metric. This encoding is realized in terms of the properties of certain well defined geometrical obje...

The general relativity theory is redefined equivalently in almost Kähler variables: symplectic form, θ[g], and canonical symplectic connection, D̂[g] (distorted from the Levi–Civita connection by a tensor constructed only from metric coefficients and their derivatives). The fundamental geometric and physical objects are uniquely determined in metric compatible form by a (pseudo) Riemannian metri...

Manifold Learning learns a low-dimensional embedding of the latent manifold. In this report, we give the definition of distance metric learning, provide the categorization of manifold learning, and describe the essential connection between manifold learning and distance metric learning, with special emphasis on nonlinear manifold learning, including ISOMAP, Laplacian Eigenamp (LE), and Locally ...

Let (M, I) be an almost complex 6-manifold. The obstruction to integrability of almost complex structure (socalled Nijenhuis tensor) N : Λ0,1(M)−→ Λ(M) maps a 3-dimensional bundle to a 3-dimensional one. We say that Nijenhuis tensor is non-degenerate if it is an isomorphism. An almost complex manifold (M, I) is called nearly Kähler if it admits a Hermitian form ω such that ∇(ω) is totally antis...

Ricci solitons (RS) have an extensive background in modern physics and are extensively used cosmology general relativity. The focus of this work is to investigate almost (RAS) on Lorentzian manifolds with a special metric connection called semi-symmetric u-connection (SSM-connection). First, we show that any quasi-Einstein manifold having SSM-connection, whose RS, Einstein manifold. A similar c...

In this paper, we initiate the study of conformal $\eta$-Ricci soliton and almost within framework para-Sasakian manifold. We prove that if metric admits soliton, then manifold is $\eta$-Einstein either vector field $V$ Killing or it leaves $\phi$ invariant. Here, have shown characteristics scalar curvature when admitting pointwise collinear with characteristic $\xi$. Next, show a endowed an in...

We study balanced Hermitian structures on almost abelian Lie algebras, i.e. algebras with a codimension-one ideal. In particular, we classify six-dimensional which carry structure. It has been conjectured in [1] that compact complex manifold admitting both metric and an SKT necessarily Kähler metric: prove this conjecture for solvmanifolds left-invariant structures. Moreover, investigate the be...

In this paper, we study the generalized m-quasi-Einstein metric in context of contact geometry. First, prove if an H-contact manifold admits a with non-zero potential vector field V collinear ?, then M is K-contact and ?-Einstein. Moreover, it also true when H-contactness replaced by completeness under certain conditions. Next, that complete closed whose thenMis compact, Einstein Sasakian. Fina...